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Question #41559

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the region enclosed by the parametric curve $x(t) = t^3 - 3t$ and $y(t) = (t^2 - 1)^2 (t^2 - 3)$ , where the parameter $t$ satisfies the inequality $(t^2 - 1)^2 (t^2 - 3) \le 0$ . The area of the region $R$ is

Options

A.
$\frac{96\sqrt{3}}{35}$
B.
$\frac{1664 - 288\sqrt{3}}{105}$
C.
$\frac{832}{105}$
D.
$\frac{32\sqrt{3}}{35}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Bounded vs Unbounded Regions|Parametric Curve Area|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of areas bounded by parametric curves and the interpretation of inequalities with even multiplicity roots.

A common trap is assuming that the even multiplicity roots ( $t = \pm 1$ ) divide the region into separate loops that must be integrated with absolute values.

However, these roots merely represent points where the curve is tangent to the x-axis, and the entire interval $[-\sqrt{3}, \sqrt{3}]$ traces a single, simple closed curve.

Splitting the integral at $t = \pm 1$ and summing absolute values incorrectly calculates overlapping areas, leading to the trap option.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Bounded vs Unbounded Regions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free